Weak Hopf Algebra Structure Of The Weak Quantum Algebra

Quantized enveloping algebras for Kac-Moody algebras were introduced indepen-dently by Dirfeld [D1] and Jimbo [Ji] in studying the quantum Yang-Baxter equation andtwo-dimensional solvable lattice modules. In 1988, Borcherds [B] introduced generalizedKac-Moody algebras to accommodate his study of monstrous moonshine and the vertexrepresentation of the monster simple group. The structure and the representation theoryof generalized Kac-Moody algebras are very similar to those of Kac-Moody algebras, anda lot of facts about Kac-Moody algebras can be extended to generalized Kac-Moody al-gebras without difficulty. But there are some differences, too. For example, generalizedKac-Moody algebras may have imaginary simple roots whose multiplicity can be greaterthan 1. Also, they may have infinitely many simple roots. In 1995, Kang [Kn] had con-structed quantum deformations for generalized Kac-Moody algebras and their modules.Lie superalgrbras can be regarded as a generalization of generalized Kac-Moody algebras.For Borcherds superalgebras defined by a symmetrizable Cartan matrix, Georgia Benkart,Seok-Jin Kang and Melville [BKM] have described quantized enveloping algebras andgiven an explicit expression for their Verma modules, and their irreducible highest weightmodules.On the other hand, because of the introduction of quantum groups [D2], the im-portance of Hopf algebras has been widely recognized in both mathematics and physics.Recently, generalization of Hopf algebras has been considered. A well-known example isweak Hopf algebra, which is introduced in [Lil]. A bialgebra H over a field k is calleda weak Hopf algebra if there exists T∈Hom_k(H, H) such that T*id*T=T andid*T*id=id, where*is the convolution product; T is called a weak antipode of H.In present paper, a weak Hopf algebra always in this sense. As is known, two typicalexamples of such weak Hopf algebras have been found, which are the monoid algebra kS[Lil] of a regular monoid S and the weak quantum algebra wsl_q(2) and vsl_q(2) [li2] (seealso [Ai] for weak Hopf algebras corresponding to msl_q~d(n). In 2005, Yang has given amore nontrivial weak Hopf algebra m_q~d((?)) [Y] associated the semisimple Lie algebra (?).Following the idea of [Y], we will construct three more general weak quantum alge-bras wV_q((?)), wU_q~d((?)) and n_q~d((?)). Then we discuss the structure and some presentationsof them. When we try to generalize Yang's result to the generalized Kac-Moody algebraand Borcherds superalgebra, we must deal with the image roots andθ-colored matrix. The method is not trivial. Moreover, this paper not only give some new weak Hopf algebras, but also give new method to study generalized Kac-Moody algebras and Borcherdssuperalgebras. The detailed outline of this paper is as follows.In Chapter 1 we define a special quantum enveloping algebra V_q(G) associated thesemisimple Lie algebra G. Then we construct the corresponding weak quantum algebrawV_q(G), which has a weak Hopf algebra structure. We will define the highest weightmodule and Verma module over the algebra wV_q(G). Moreover, we study the isomorphismamong wV_p(G) and wV_q(G).Chapter 2 constructs a weak quantum algebra wU_q~d(G) associated the generalizedKac-Moody algebra G. We prove wU_q~d(G) is a weak Hopf algebra. We also discuss thebasis of wU_q~d(G). Then the highest weight module and the weak A—-form of wU_q~d(G) arestudied. In the finial section, we study the subalgebra wU_i~d of rank 1. The center of wU_i~dis constructed.In Chapter 3, we define a more general weak quantum algebra n_q~d(G), which G isa Borcherds superalgebra. Then we introduce the definition of weak Hopf superalgebra,which is similar to the definition of weak Hopf algebra in the sense of Li. A weak Hopfsuperalgebra is a bisuperalgebra (H,μ,η,Δ,ε) equipped with aθ—colored algebra antimorphism T: Hâ†'H such that T*id*T=T and id*T*id=id, where T is calleda weak antipode of H. We prove n_q~d(G) is a weak Hopf superalgebra. In particular, if Gis a semi-simple Lie algebra, then n_q~d(G) is just m_q~d(G) in [Y]; if G is a special generalizedgac-Moody algebra which satisfies m_i=1 for all i∈I, then n_q~d(G) is just wU_q~d(G) inChapter 2. Finally, we define the weak A-form wU_A~d, and discuss the property of wU_A~dunder the limit of qâ†'1.Throughout the paper, we assume the basic field is k whose character is 0. Allalgebras, modules and vector spaces are over k without specified.